Mathematical Education Notes

Edited by John R. Mayor, AAAS and University of Maryland
Collaborating Editor: John A. Brown, University of Delaware

Activity and Motivation in Mathematics

Edwin E. Moise, Harvard University

It often happens that when a committee meets to discuss a question of educational policy, the mere fact that the committee exists and is holding a meeting tends to suggest a commitment on the question under discussion. For example, at a meeting of a Committee on the Teaching of Calculus to Engineers, it is hard to avoid the suggestion that the teaching of calculus to engineers is a special problem, requiring a special solution. This in itself is a decision. (In my own opinion, it is a wrong decision.)

It seems to me that we need to be on guard against such suggestions. And I believe that one such suggestion is present at any conference devoted to curricular planning in the large. If we look at a four-year college program, as a whole, it is very natural to judge it by the quantity of mathematical knowledge that it appears to contain, and by the speed with which it reaches the concepts and methods of contemporary mathematics. Surely these are important criteria. I believe, however, that there are other criteria which it is easy to neglect, because they are harder to describe and to apply.

In the first place, I do not believe the effect of an educational program can safely be judged merely by the quantity of knowledge which the student can demonstrate on an examination. It seems to me that this sort of simple bookkeeping is not just slightly inaccurate but grossly so. An extreme example may make this point plain.

I worked for the doctorate in mathematics under Professor R.L. Moore, of The University of Texas. In three and a half years of graduate work, under his direction, I believe that I heard him lecture (in a style that a conventional professor would recognize as lecturing) for about two hours at most, and it may well have been less. All of his graduate courses worked in the same way: he would present postulates, definitions, and propositions; and the student's job was to find out which of the propositions were true, and present proofs or counter-examples. The theorems were not mere exercises; they were the substance of the sort of set-theoretic topology that Moore was teaching. Six months might elapse, between the time a problem was proposed and the time it was solved. At first, a very large number of the students' proofs were wrong. When a student had gone astray, Moore did not correct him; he would merely say that he didn't understand; and it took us quite a while to realize that anything that he "didn't understand" was surely wrong.

Under Moore's regime, all use of the literature is forbidden. Most of the time, therefore, a student of Moore doesn't even know whether the problems that he is working on are research problems. Eventually, they turn out to be, and the result is a thesis. Thus, at the time when I wrote my first research paper, I had never read one. And I had never discussed topology with fellow-students outside of class.

If this scheme of teaching seems misguided, we should remember that the list of Moore's students includes the names of R.L. Wilder, G.T. Whyburn, R.H. Bing, and many other research mathematicians of distinguished achievements. (I could give a long list, but would prefer to avoid the problem of deciding where to stop.) And Moore's record as a teacher is even more impressive in the light of the circumstances under which he has taught. There are some universities in the United States where a professor can take for granted that the best and most ambitious graduate students in the country will arrive, every year, already well trained and already fully committed as mathematicians. The University of Texas is not one of these, and so Moore has had the task of recruitment as well as the task of teaching. When he first encountered them, R.L. Wilder intended to be an actuary, and G.T. Whyburn intended to be a chemist. Bing made his first appearance as a summer student, on vacation from his regular job as a high-school teacher.

Thus Moore's teaching has been effective, to say the least. I believe that all teaching would profit if the basis of his method were better understood. By this I do not mean to suggest that his method can simply be copied, in the extreme form in which he practices it. It seems rather likely that the success of the method, in its extreme form, depends on special conditions.

First, the method may require that the teacher be a genius. Teaching in this style is not merely a negative matter of not lecturing; rather, it is a striking example of the art that conceals art, and it makes great demands on both mathematical and psychological depth.

Second, it may require that the teacher dominate the environment that his students live in. Surely Moore has done this at Texas, but I doubt that he or anybody else could do it, at Harvard, Princeton, or Chicago, well enough to persuade students not to talk to each other, or well enough to persuade other professors to avoid telling them things that would help in their problems.

Third, the method may require that the subject-matter be logically primitive, and fairly isolated from that of other courses.

In spite of all these reservations, however, I believe that Moore's work proves something of broad significance. It proves that sheer knowledge does not play the crucial role in mathematical development that most people suppose. The amount of knowledge that a small class can acquire, struggling at every stage to produce its own proofs, is quite small. The resulting ignorance ought to be a hopeless handicap, but in fact it isn't; and the only way that I can see to resolve this paradox is to conclude that mathematics is capable of being learned as an activity, and that knowledge which is acquired in this way has a power which is out of all proportion to its quantity.

For this reason, it seems to me quite unrealistic to judge a curriculum by its general outline, or to judge a course by its syllabus. We can "cover" very impressive material, if we are willing to turn the student into a spectator. But if you cast the student in a passive role, then saying that he has "studied" your course may mean no more than saying of a cat that he has looked at a king. Mathematics is something that one does. In elementary courses, this idea is well understood; but later it often gets lost, only to reappear, as a brutal surprise, when the time comes to write a dissertation. When this happens, a high price has been paid for erudition: we seem to be relying on the idea that the creative spirit is both self-discovered and indestructible. This is like the idea that "murder will out": if it were true, how would anybody know it?

It is, I believe, a fundamental task of the teacher to introduce students to the intellectual life that he, the teacher, really leads. Under this standard, the cram-school conception of learning is a falsification of the subject. And the subject can be falsified in other ways, by presenting ideas in advance of their motivations. This can be explained best by a few examples.

1. A distinguished algebraist once served as an examiner in a final oral for the doctorate, based on a dissertation on Banach algebras. Toward the end of the examination, the algebraist asked the student to describe some examples Of Banach algebras. The student was able, at length, to name one example, but his one example was trivial.

The director of the dissertation could have given a serious answer to the question; he knew of good reasons for investigating Banach algebras. But apparently his teaching had taken a shortcut, past "classical mathematics" to "modern mathematics," and the result was that the student was working on Banach algebras for no good reason known to himself. Thus the student's training omitted an essential part of the teacher's intellectual equipment.

2. In a course in analysis, the students were furnished with a very elegant definition of a Riemann surface. One of them showed the definition to a personal friend who was an analyst and a member of the National Academy of Science. It took the academician quite a while to figure out how the definition worked.

To the teacher in this course, the "right" definition of a Riemann surface was the solution of a difficult problem. But it is hard to understand the answer if you don't know what the question was.

3. One of the veterans of Bourbaki once told me that he had been annoyed by people who approached him at meetings and told him about trivial syntactical tricks which they had devised. These people expected Bourbaki to admire their tricks, but had the bad luck to try them on a man who was distinguished (even among the Bourbakistes) for his candor. It seems that the source of this misunderstanding is that while the Bourbaki series is self-contained in a logical sense, it is not psychologically self-contained; the first few volumes, taken at face value by a naive reader, give a misleading impression of the mentalities of the authors. The authors are learned and tough-minded; and some of the simplest looking things, in the first few volumes, are the solutions of difficult problems. (Where do you begin? and how? Above all, what do you omit?)

This is not, in itself, an example of the sort of misguided teaching that I am talking about. (The Bourbaki books are not courses, and I hardly think that courses taught by Bourbakistes have such an effect.) With this reservation, the preceding three examples have something in common. Each of them is an example of first-rate mathematics presented apart from the context which makes it significant; in each case, the solution was presented before the problem.

The point, I believe, is that mathematics is arranged not only in a logical order but also in a psychological order; and at many stages, the orders are different. Thus in algebra, the abstract and general logically precede the concrete and special, but most of the time the abstractions are intended to be codifications, and may easily seem pointless to a student who doesn't know what is being codified.

These remarks are not intended as a defense of "traditional" teaching. Mathematics has no traditions comparable to those of literature and art: some of the best of its past is dead, at least in a stylistic sense. Thus the task of a calculus student is to understand calculus, and for this it is neither necessary nor sufficient that he understand Newton (let alone Leibniz). The problems that I have been discussing would not even arise if we were not reformulating the curriculum; and they are important precisely because such reformulations are necessary.

The problem of arranging a curriculum in an order which is logically sensible and psychologically well-motivated appears to be highly empirical. As far as I know, there are no general principles which can be relied upon. Lately, some authors have proposed the so-called genetic principle, under which mathematics should be presented in the order in which it was discovered. But this involves difficulties. Manipulative algebra, and negative real numbers, were developed during the Renaissance. Therefore, under the genetic principle, you should teach the geometry of Euclid (including the Eudoxian theory of proportionality, for incommensurables) before you teach the student to write or solve the equation 2x+6=0. In fact, you should teach Euclid before you teach the representation of positive integers in decimal form. No one seriously proposes to do this. But a rule which admits massive exceptions is not a very reliable rule. Indeed, to rescue the genetic principle from absurdity, we must construe it so loosely that it can hardly be used at all. It is not even safe, as far as it goes. If you are teaching, say, Eighteenth Century mathematics, in a given year, then you cannot assume that the student knows all the mathematics of earlier centuries, because in fact you have taught him only a small portion of it. You must make sure, in each case, that what you are teaching is not based (logically or psychologically) on an appeal to earlier mathematics which has not been taught.

For these reasons, I believe that the people who have proposed the genetic principle were really thinking about the problem of motivation. In practice, most unmotivated treatments are violations of it. But in a serious attack on the problems of course design, the genetic principle ought to be regarded merely as a fund of suggestions, and not as a standard of judgment.

We also hear the maxim that the particular should precede the general. Usually this gives reasonable results, but here again there are exceptions. The easiest way that I know of, to see that the number 5 has a cube root, is to observe (somehow) that the cube function is continuous, and so must take on every value between O and 8. I see no advantage in specializing the mean-value theorem to which this argument appeals; its natural habitat, both logically and psychologically, is the set of functions continuous on intervals.

For these reasons, I believe that there are no simple answers to the questions that I have been raising. The fundamental task of a mathematician who teaches is to convey to his students not only what mathematicians know, but also what they do, and how, and why. This is a problem in full and honest self-revelation. And (as Raymond Chandler has remarked in another connection) honesty is an art.

A revised version of an address delivered at a conference on the teaching of college mathematics, in Katada, Japan, Sept, 6, 1964.